The homotopy groups of the sphere spectrum are the stable homotopy groups of spheres.

However, could you help me to appreciate the mathematical meanings of the following:

What is the significance of the meanings of "Anderson dual" to the sphere spectrum? (For example, I understand the significant meanings of Pontryagin dual and torsion subgroup. Can "Anderson dual" be explained as simple as that of Pontryagin dual?)

What is the significance of the meanings of (a shift of) $I\mathbb{C}^{\times}$, a “character dual” to the sphere spectrum?

Freed also states that Anderson duality is the analog of the integral dual (which takes $A$ to Ext$^{\cdot}(A, \mathbb{Z})$, this operation being a derived improvement of the naive integral dual that takes $A$ to Hom$(A,\mathbb{Z})$).
– j.c.Oct 12 at 1:29

2 Answers
2

The Anderson dualizing spectrum $I_\mathbf{Z}$ can be defined as follows. Consider the functor $X\mapsto \mathrm{Hom}(\pi_{-\ast} X,\mathbf{Q/Z})$ from the homotopy category of spectra to graded abelian groups. Since $\mathbf{Q/Z}$ is an injective $\mathbf{Z}$-module, this functor is representable by a spectrum $I_\mathbf{Q/Z}$, called the Brown-Comenetz dualizing spectrum. After $p$-completion, this spectrum is equivalent to the spectrum you denoted $I\mathbf{C}^\times$. If you do the same construction with $\mathbf{Q/Z}$ replaced by $\mathbf{Q}$, you obtain the Eilenberg-Maclane spectrum $\mathbf{Q}$. There is a canonical map $\mathbf{Q}\to I_\mathbf{Q/Z}$, and the fiber of this map is defined to be $I_\mathbf{Z}$. This has the property that it is $(-1)$-coconnective, $\pi_0 I_\mathbf{Z} \cong \mathbf{Z}$, $\pi_{-1} I_\mathbf{Z} = 0$, and $\pi_{-n} I_\mathbf{Z} = \mathrm{Hom}(\pi_{1-n} S, \mathbf{Q/Z})$.

There are several reasons why the Anderson dualizing spectrum is significant. I'll try to explain some of them, in the order in which I understand them the best.

It is the dualizing sheaf of the affine derived scheme $\mathrm{Spec}(S)$. In order for this to make sense, I have to tell you what "dualizing sheaf" means. Let $A$ be a connective$\mathbf{E}_\infty$-ring (this definition fails drastically in the nonconnective setting). Then, an $A$-module $\omega_A$ is said to be a dualizing sheaf if:

$\omega_A$ is coconnective,

$\pi_n \omega_A$ is a finitely generated $\pi_0 A$-module for every $n$,

the map $A\to \mathrm{Map}_A(\omega_A, \omega_A)$ is an equivalence (this is the dualizing property), and

$\omega_A$ has finite injective dimension as an $A$-module, i.e., there is some integer $n$ such that for every discrete $A$-module $M$, the groups $\pi_k\mathrm{Map}_A(M,\omega_A)$ vanish for $k>n$.

You can check that $I_\mathbf{Z}$ satisfies all of these properties when $A = S$. Moreover, one can prove (see Proposition 6.6.2.1 of Lurie's SAG) that dualizing sheaves are unique up to elements of $\mathrm{Pic}(A)$; when $A = S$, this group is just $\mathbf{Z}$, generated by $S^1$. It follows that any other dualizing sheaf for $S$ is equivalent to a suspension of $I_\mathbf{Z}$.

There is an analogue of Serre duality in this spectral setting. Let me first explain the even periodic case, and then discuss what happens in the $K(n)$-local setting. In the even periodic setting, one might consider objects like $KU$, $KO$, and $\mathrm{TMF}$ and its variants. All of these can be obtained as the global sections of sheaves of $\mathbf{E}_\infty$-rings on Deligne-Mumford stacks ($\mathrm{Spec}(\mathbf{Z})$, $\mathrm{Spec}/\!\!/C_2$, and the moduli stack $\mathcal{M}_{\mathrm{ell}}$ of elliptic curves, respectively). Each of these Deligne-Mumford stacks are --- in the classical world --- smooth. In particular, they are Gorenstein.

One can actually show that the associated derived stacks are Gorenstein in the spectral world, too. (The cases mentioned above are due to Heard-Stojanoska, and Stojanoska, respectively.) One finds the extremely interesting statement, e.g., that $I_\mathbf{Z} \mathrm{TMF} := \mathrm{Map}(\mathrm{TMF}, I_\mathbf{Z})$ is in the Picard group of $\mathrm{TMF}$ (which is known to be cyclic, by work of Mathew-Stojanoska). This can be generalized: one can prove that most reasonable derived locally even periodic Deligne-Mumford stacks $\mathfrak{X}$ satisfy the property that $\mathrm{Map}(\Gamma(\mathfrak{X}, \mathcal{O}_\mathfrak{X}), I_\mathbf{Z})$ is in the Picard group of $\Gamma(\mathfrak{X}, \mathcal{O}_\mathfrak{X})$. I have a proof of the latter fact here, as Theorem 3.14; I'm sure experts already know of this result.

One practical consequence of this is the following. For a spectrum $A$, define $I_\mathbf{Z} A = \mathrm{Map}(A,I_\mathbf{Z})$. Then there is a short exact sequence
$$0\to \mathrm{Ext}^1_\mathbf{Z}(\pi_{-d-1} X, \mathbf{Z}) \to \pi_d I_\mathbf{Z} X \to \mathrm{Hom}_\mathbf{Z}(\pi_{-d} X, \mathbf{Z}) \to 0.$$
Now, suppose that $E$ is a spectrum such that $I_\mathbf{Z} E \simeq \Sigma^k E$ for some $k$ (so that $I_\mathbf{Z} E \in \mathrm{Pic}(E)$). Suppose we want to understand the $E$-theory of a spectrum $X$. Apply the short exact sequence above to the spectrum $E\wedge X$; this gives a short exact sequence
$$0\to \mathrm{Ext}^1_\mathbf{Z}(E_{-d-1} X, \mathbf{Z}) \to (I_\mathbf{Z} E)^d X \to \mathrm{Hom}_\mathbf{Z}(E_{-d} X, \mathbf{Z}) \to 0.$$
But the middle term is, by the "Anderson self-duality" of $E$, equivalent to $E^{d+k} X$. This gives an interesting and useful "universal coefficients" exact sequence.

Another reason one might care about the spectrum $I_\mathbf{Z}$ is because of its appearance in the Picard group of the $K(n)$-local category, which goes by the name Gross-Hopkins duality. Let $n>0$. After $K(n)$-localizing, the difference between $I_\mathbf{Q/Z}$ and $I_\mathbf{Z}$ vanishes: $L_{K(n)} I_\mathbf{Q/Z} \simeq \Sigma L_{K(n)} I_\mathbf{Z}$. Suppose $X$ is an $E_n$-local spectrum. We define another spectrum $I_n X = \Sigma L_{K(n)} I_\mathbf{Z} X \simeq L_{K(n)} \mathrm{Map}(L_n X, I_{\mathbf{Q/Z}})$, and let $I_n = I_n L_{K(n)} S$. Then, Gross and Hopkins proved two remarkable statements: first, the spectrum $I_n$ is invertible in the $K(n)$-local category (this is not very surprising), and second, there is an isomorphism $(E_n)^\vee_\ast I_n \simeq \Sigma^{n^2-n} (E_n)_\ast[\det]$ of $(E_n)_\ast[\![\Gamma_n]\!]$-modules, where $\det:\Gamma_n \to \mathbf{Z}_p^\times$ is the determinant map, and $\Gamma_n$ is the Morava stabilizer group at height $n$. (I've implicitly been fixing the Honda formal group over $\mathbf{F}_{p^n}$.)

The proof of this result is really cool; it passes through an important object in rigid analytic geometry (the Gross-Hopkins period map). In any case, if $p\gg n$, then one can actually identify $I_n$ itself with a $K(n)$-local object $S[\det]$, called the determinantal sphere. This object is incredibly interesting, and plays an important role in the chromatic story at height $n$.

The Anderson dualizing spectrum also plays an important (but unpublished) role in orientation theory. The following simple question is an open problem: let $\kappa$ be a perfect field of characteristic $p>0$, and let $H$ be a formal group of finite height over $\kappa$. Denote by $E(\kappa, H)$ the associated Morava $E$-theory. Then, is there an $\mathbf{E}_\infty$-orientation $MU \to E(\kappa,H)$? One can reduce this to understanding the spectrum of units $gl_1 E(\kappa,H)$. In unpublished work, Hopkins and Lurie have shown that the fiber of the map $gl_1 E(\kappa, H) \to L_n gl_1 E(\kappa, H)$ (which is known as the discrepancy spectrum) is equivalent to $\Sigma^n I_\mathbf{Q/Z}$ in nonnegative degrees (they also did more). This allows one to construct a map $\Sigma^{n+1} H\mathbf{Z} \to gl_1 E(\kappa, H)$, the very existence of which is enough to obstruct the existence of certain $\mathbf{E}_\infty$-orientations of $E(\kappa, H)$.

I have heard that there is some relationship with TQFTs, but I don't know how that goes. There are people here more qualified than me who can say more about this.

Re: a relationship with TQFTs: groups of isomorphism classes of invertible TQFTs can be calculated as groups of homotopy classes of maps of certain spectra. The domain spectrum is a Madsen-Tillmann spectrum, and the codomain spectrum depends on your choice of target $n$-category. In low dimensions, natural choices of target categories give us for codomain spectra the connective covers of $\Sigma^n I\mathbb C^\times$ (for isomorphism classes of invertible TQFTs) and $\Sigma^{n+1} I\mathbb Z$ (for deformation classes). Conjecturally this is true in all dimensions.
– Arun DebrayOct 12 at 1:41

Thanks, @ArunDebray! I recall we had a discussion about this a long time back.
– skdOct 12 at 2:12

One way to think of these spectra is in terms of the cohomology theories they define. In other words, if
$E$ is a spectrum, what is $[E, I\mathbb Z]$? This is less of a description of what they are and more of a
description of what they do; there are probably other, more conceptual answers to your question.

1. $I\mathbb Z$ and $\Sigma^n I\mathbb Z$

The universal coefficient theorem describes how to compute cohomology groups from homology groups: there is a short
exact sequence

Nonetheless, part of the story can be salvaged: if you try this with stable homotopy groups (the homology theory
represented by the sphere spectrum), you obtain the cohomology theory represented by the Anderson dual of the
sphere. That is, for any spectrum $X$ there is a short exact sequence
$$ 0\longrightarrow \mathrm{Ext}^1(\pi_{n-1}(X), \mathbb Z)\longrightarrow [X, \Sigma^n I\mathbb Z]\longrightarrow
\mathrm{Hom}(\pi_n(X), \mathbb Z)\longrightarrow 0,$$
and it splits noncanonically. (See Freed-Hopkins, §5.3.)

These spectra provide an analogue of Pontrjagin duality. If $A$ is an abelian group, the set of maps $A\to\mathbb
C^\times$ is an abelian group under pointwise multiplication, and this is called the Pontrjagin dual of $A$. An
analogue for spectra might be the assignment $X\mapsto \mathrm{Hom}(\pi_nX, \mathbb C^\times)$, the group of
characters of the $n$th homotopy group of $X$. This is precisely what $[\Sigma^n X, I\mathbb C^\times]$
is; more broadly, one could describe $I\mathbb C^\times$ as the spectrum whose cohomology theory is calculated by
$$(I\mathbb C^\times)^n(X) = \mathrm{Hom}(\pi_{-n}(X), \mathbb C^\times).$$

As an addendum, I'm guessing this question arose because of the appearance of these spectra in physics,
specifically in the classification of invertible topological field theories. Following Freed-Hopkins, the
classification of invertible TQFTs $\mathsf{Bord}_n\to \mathsf C$, where $\mathsf C$ is some target symmetric
monoidal $(\infty, n)$-category, is equivalent to the abelian group of homotopy classes of maps between the
classifying spectrum of the groupoid completion of $\mathsf{Bord}_n$ and the classifying spectrum of the groupoid
of units of $\mathsf C$. The classifying spectrum of $\mathsf{Bord}_n$ is determined by
Schommer-Pries, and for certain reasonable choices of $\mathsf C$, we get
$\Sigma^nI\mathbb C^\times$ and $\Sigma^{n+1} I\mathbb Z$ (for small $n$, and conjecturally for all $n$).

For small $n$, we know some good chocies for $\mathsf C$: for example, if we let $n = 1$, we can take $\mathsf C$
to be the category of super-vector spaces, and for $n = 2$ we can take the Morita 2-category of superalgebras. In
both cases, the classifying spectrum is the connective cover of $\Sigma^n I\mathbb C^\times$. This suggests that in
higher dimensions $n$, we might find symmetric monoidal $n$-categories $\mathsf C$ whose classifying spectra
continue this pattern, seeing more and more of $I\mathbb C^\times$, and the calculations of SPT phases coming out
of physics provide heuristic evidence for this conjecture.

This used the discrete topology on $\mathbb C$. If you instead give $\mathbb C$ the usual topology when defining the
categories of super-vector spaces or superalgebras, you get different classifying spectra: the connective covers of $\Sigma^{n+1} I\mathbb Z$ for $n =
1$, resp. $2$. Conjecturally, this pattern also continues further. Using the usual topology on $\mathbb C$ corresponds to classifying
deformation classes of invertible TQFTs rather than isomorphism, and again physics calculations provide some evidence for the conjecture.

Thanks, when I see "a shift" in the "a shift of" the Anderson dual Iℤ$_1$ to the sphere spectrum -- what does "a shift" really mean? Is this a standard terminology?
– wonderichOct 12 at 2:50

1

"Shift" here means suspension, and is standard terminology. The idea is that suspension shifts the degrees of the stable homotopy groups: if $X$ is a spectrum, $\pi_n(X) \cong \pi_{n+1}(\Sigma X)$. You therefore can think of it as "shifting" the entire spectrum. This terminology may also have come from triangulated category theory: a triangulated category includes data of a something called a shift functor, usually denoted $[1]$, and the homotopy category of spectra is a triangulated category, with $[1] = \Sigma$.
– Arun DebrayOct 12 at 2:54